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AS 3.15 Simultaneous Equations Starters

This resource provides practice exercises for solving 3D simultaneous equations and explores the geometric interpretations of the equations in 3D planes. It also includes applications of solving equations in parallel planes and with rearrangement, and addresses consistent and inconsistent situations.

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AS 3.15 Simultaneous Equations Starters

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  1. 2 variables AAns 2 variables BAns Introduce 3D planes CAns 3D plane practice DAns Solving 3D Simultaneous EAns Solving with negatives FAns Solving with rearrangement GAns An Application HAns Another Application IAns Consistent Situations JAns More Consistency KAns Inconsistent LAns More Inconsistent MAns AS 3.15 Simultaneous Equations Starters

  2. Simultaneous Eqns A • Sketch the graphs of • Y = 2x – 4 • y = -0.5x + 6 • Where do the two lines • intersect? • 4) Write an equation of a • line parallel to line 1)

  3. Simultaneous Eqns A ans • Sketch the graphs of • Y = 2x – 4 • y = -0.5x + 6 • Where do the two lines • intersect? • 4) Write an equation of a • line parallel to line 1)

  4. Simultaneous Eqns B • Solve simultaneous equations • y = 2x – 4 • y = -0.5x + 6 • 2) 2x + 3y = 18 • x – 2y = -5 • Find the ‘x’ and the ‘y’ intercepts of the equations in question 2) above • Rearrange the equations in Q 1) above to be in the form ax + by = c

  5. Simultaneous Eqns B ans • Solve simultaneous equations • y = 2x – 4 • y = -0.5x + 6 • 2) 2x + 3y = 18 • x – 2y = -5 • Find the ‘x’ and the ‘y’ intercepts of the equations in question 2) above • Rearrange the equations in Q 1) above to be in the form ax + by = c

  6. Simultaneous Eqns C • Given two equations with two variables, explain geometrically the three possible situations that can • (in a 3D situation) Describe the plane z = 3 • (in a 3D situation) Describe the plane x = -5 • Find the three axis intercepts for the plane • 2x + 3y – z = 12 • Sketch the plane (in 3D) x + 2y = 8

  7. Simultaneous Eqns C ans • Given two equations with two variables, explain geometrically the three possible situations that can • (in a 3D situation) Describe the plane z = 3 • (in a 3D situation) Describe the plane x = -5 • Find the three axis intercepts for the plane • 2x + 3y – z = 12 • Sketch the plane (in 3D) x + 2y = 8

  8. Simultaneous Eqns D • 1) Find the three axis intercepts for the plane • x + 2y + 3z = 9 • Write an equation for a plane parallel to the plane in 1) • 3) Find the three axis intercepts for the plane • x – 4z = 20 • 4) What effect would changing the coefficient of ‘y’ have on the plane in question 1? (From 2y to 9y) • 5) Find the three axis intercepts for the plane • 3x + 15 = 2y

  9. Simultaneous Eqns D ans • 1) Find the three axis intercepts for the plane • x + 2y + 3z = 9 • Write an equation for a plane parallel to the plane in 1) • 3) Find the three axis intercepts for the plane • x – 4z = 20 • 4) What effect would changing the coefficient of ‘y’ have on the plane in question 1? (From 2y to 9y) • 5) Find the three axis intercepts for the plane • 3x + 15 = 2y

  10. Simultaneous Eqns E • 1) Solve these simultaneous Equations • x + 2y – z = 2 • x + y + z = 6 • 2x + y – z = 1 • Write an equation for a plane parallel to the first plane above • Find the three intercepts to the first equation above

  11. Simultaneous Eqns E ans • 1) Solve these simultaneous Equations • x + 2y – z = 2 • x + y + z = 6 • 2x + y – z = 1 • Write an equation for a plane parallel to the first plane above • Find the three intercepts to the first equation above

  12. Simultaneous Eqns F • 1) Solve these simultaneous Equations • 3x – 2y – 2z = -6 • 4x – y – 2z = -1 • 3x + 4y + z = 21 • Write a different equation for a plane identical to the first plane above. • Describe geometrically what is happening in Question 1

  13. Simultaneous Eqns F ans • 1) Solve these simultaneous Equations • 3x – 2y – 2z = -6 • 4x – y – 2z = -1 • 3x + 4y + z = 21 • Write a different equation for a plane identical to the first plane above. • Describe geometrically what is happening in Question 1

  14. Simultaneous Eqns G • 1) Solve these simultaneous Equations • x = y + z + 4 • 2x + 4y = 15 + 3z • x + y = 5 • Write an equation for a plane parallel to the second plane above. • Describe features of the plane formed by the first equation above.

  15. Simultaneous Eqns G ans • 1) Solve these simultaneous Equations • x = y + z + 4 • 2x + 4y = 15 + 3z • x + y = 5 • Write an equation for a plane parallel to the second plane above. • Describe features of the plane formed by the first equation above.

  16. Simultaneous Eqns H By some miracle, Zoe can achieve her daily brain endorphin hit by solving three different types of mathematical problems: algebra, binary, and calculus Each algebra problem takes 1 hour to do, involves 1 session of tears, and fills 2 pages in her book Each binary problem takes 3 hours to do, involves 1 session of tears, and fills 1 page in her book Each calculus problem takes 2 hours to do, also involves 1 session of tears, and fills 4 pages in her book One eventful evening she spends 8 hours on maths problems, has only 4 sessions of tears, and fills 11 pages in her book. How many of each type of problem did she do? (at Achieve level, cos the Merit is like this problem – too hard)

  17. Simultaneous Eqns H ans How many of each type of problem did she do?

  18. Simultaneous Eqns I Riley over-stepped the mark on yesterdays starter problem, so as punishment had to do exactly 80 press-ups, make up 68 worksheets and 24 starter problems of an acceptable standard. On a school day he could do 1 press-ups, 2 work sheets and 1 starter problem. On a weekend day he could do 2 press-ups, 3 work sheets and 1 starter problem. On a holiday day he could do 5 press-ups, 3 work sheets and 1 starter problem. How many weekend days will Riley need to be punished for?

  19. Simultaneous Eqns I ans How many weekend days will Riley need to be punished for? Press-ups 1Sc + 2w + 5h = 80 Worksheets 2Sc + 3w + 3h = 68 Starters 1Sc + 1w + 1h = 24

  20. Simultaneous Eqns J • 1) Draw and explain all the situations where 3 variable simultaneous are ‘consistent’ - ie have solution(s) • Write down a set of 3 variable simultaneous equations that could give each of the situations above

  21. Simultaneous Eqns J ans • 1) Draw and explain all the situations where 3 variable simultaneous are ‘consistent’ - ie have solution(s) • Write down a set of 3 variable simultaneous equations that could give each of the situations above

  22. Simultaneous Eqns K • Solve this system of equations: • 4x + 3y + 4z = 29 • 2x = 3z • 3x + z = 5 + 2y • Given that this system of equations • x + 2y – z = 2 • x + y + z = 6 • ? • Form a third equation to make a system with infinite solutions • 3) Form another third equation to make a different situation that has infinite solutions possible

  23. Simultaneous Eqns K ans • Solve this system of equations: • 4x + 3y + 4z = 29 • 2x = 3z • 3x + z = 5 + 2y • Given that this system of equations • x + 2y – z = 2 • x + y + z = 6 • ? • Form a third equation to make a system with infinite solutions • 3) Form another third equation to make a different situation that has infinite solutions possible

  24. Simultaneous Eqns L • 1) Draw and explain all the situations where 3 variable simultaneous are ‘inconsistent’ - ie have NO solutions • Write down a set of 3 variable simultaneous equations that could give each of the situations above

  25. Simultaneous Eqns L ans • 1) Draw and explain all the situations where 3 variable simultaneous are ‘inconsistent’ - ie have NO solutions • Write down a set of 3 variable simultaneous equations that could give each of the situations above

  26. Simultaneous Eqns M • Solve this system of equations: • x + 3y + 2 = 3z • 2x + 3 = y + 2z • 3x + 3 = 3z • Given that this system of equations • x + 2y – z = 2 • 2x + y + 3z = 1 • ? • Form a third equation to make a system with NO solutions • 3) Form another third equation to make a different situation that has NO solutions possible

  27. Simultaneous Eqns M ans • Solve this system of equations: • x + 3y + 2 = 3z • 2x + 3 = y + 2z • 3x + 3 = 3z • Given that this system of equations • x + 2y – z = 2 • 2x + y + 3z = 1 • ? • Form a third equation to make a system with NO solutions • 3) Form another third equation to make a different situation that has NO solutions possible

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